# Fundamental theorem of linear algebra

In mathematics, the fundamental theorem of linear algebra makes several statements regarding vector spaces. These may be stated concretely in terms of the rank r of an m × n matrix A and its singular value decomposition:

$A=U\Sigma V^\mathrm{T}\$

First, each matrix $A \in \mathbf{R}^{m \times n}$ ($A$ has $m$ rows and $n$ columns) induces four fundamental subspaces. These fundamental subspaces are:

name of subspace definition containing space dimension basis
column space, range or image $\mathrm{im}(A)$ or $\mathrm{range} (A)$ $\mathbf{R}^m$ $r$ (rank) The first $r$ columns of $U$
nullspace or kernel $\mathrm{ker}(A)$ or $\mathrm{null} (A)$ $\mathbf{R}^n$ $n - r$ (nullity) The last $(n - r)$ columns of $V$
row space or coimage $\mathrm{im}(A^\mathrm{T})$ or $\mathrm{range} (A^\mathrm{T})$ $\mathbf{R}^n$ $r$ (rank) The first $r$ columns of $V$
left nullspace or cokernel $\mathrm{ker}(A^\mathrm{T})$ or $\mathrm{null} (A^\mathrm{T})$ $\mathbf{R}^m$ $m - r$ (corank) The last $(m - r)$ columns of $U$

Secondly:

1. In $\mathbf{R}^n$, $\mathrm{ker}(A) = (\mathrm{im}(A^\mathrm{T}))^\perp$, that is, the nullspace is the orthogonal complement of the row space
2. In $\mathbf{R}^m$, $\mathrm{ker}(A^\mathrm{T}) = (\mathrm{im}(A))^\perp$, that is, the left nullspace is the orthogonal complement of the column space.
The four subspaces associated to a matrix A.

The dimensions of the subspaces are related by the rank–nullity theorem, and follow from the above theorem.

Further, all these spaces are intrinsically defined – they do not require a choice of basis – in which case one rewrites this in terms of abstract vector spaces, operators, and the dual spaces as $A\colon V \to W$ and $A^* \colon W^* \to V^*$: the kernel and image of $A^*$ are the cokernel and coimage of $A$.